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Transcript
Comment.Math.Univ.Carolinae 36,3 (1995)579{584
On resolvable spaces and groups
Luis Miguel Villegas-Silva
Abstract. It is proved that every uncountable !-bounded group and every homogeneous
space containing a convergent sequence are resolvable. We nd some conditions for
a topological group topology to be irresolvable and maximal.
Keywords: resolvable, maximal, -bounded
Classication: 20K45, 54D20, 22A05, 54H11
1. Introduction
Hewitt [H] introduced and studied the properties of being irresolvable and/or
maximal for topological spaces. He constructed the rst examples of irresolvable spaces by generating maximal topologies. Several years later Malykhin [M],
assuming some extra set-theoretic axioms, gave the rst and, as far as we know,
the only example of a countable irresolvable topological group, which is maximal
as well. This raises the question about the existence of non-maximal irresolvable groups. In the case of general topological spaces there are several examples
of such spaces ([A], [P]). This paper contributes to the problem of irresolvable
non-maximal groups by establishing conditions on the weight of maximal topological groups. The results about irresolvable groups hold under the assumption
of the combinatorial principle p = c. We also prove the resolvability of certain uncountable -bounded groups and of a homogeneous space containing convergent
sequences.
The notation and denitions used in this paper is the same as in the related
literature, but the following are specied: represents the group of nite subset
of ! with the symmetric dierence as a group operation. The symbol hH i denotes
the subgroup generated by the set H . An ordinal is the set of its predecessors.
We only consider Hausdor non-discrete topological groups and Hausdor spaces
which are dense in themselves. The symbol e denotes the identity element of
a group. For the weight and density of a topological space X we use w(X ) and
d(X ). The symbol ! is the rst innite ordinal and c is the least ordinal of the
same cardinality as fx : x !g. The closure of a subset U of X is denoted by U , .
This work was partially supported by DGAPA-UNAM, Mexico
579
580
L.M.Villegas-Silva
2. Denitions and preliminary results
This section contains the relevant concepts, denitions and results in the theory
of maximal and irresolvable spaces which form the basis of our work.
The following two denitions were introduced by E. Hewitt.
Denition 2.1. A space X is k-resolvable (2 k) if X contains k disjoint dense
subsets. If k = 2, we simply say that X is resolvable. A space which is not
resolvable is called irresolvable.
Denition 2.2. A topological space (X; ) dense in itself is maximal (and is
a maximal topology) if every dense in itself topology 0 on X with 0 coincides
with .
Denition 2.3 [G]. Let be an innite cardinal. A topological group X is
called -bounded if for every neighborhood U of the identity there exists a subset
K X with jK j such that X = K U .
It is not hard to see that every subgroup of an -bounded topological group
is -bounded. The metrizable @0 -bounded groups are separable, and every @0 bounded group may be embedded as a subgroup of a product of separable groups
([G]).
We will use the following results.
Theorem 2.4 [CF]. Let X = Si2I Xi be a space with each Xi resolvable. Then
X is resolvable. In particular, every homogeneous spaces containing a resolvable
subspace is itself resolvable.
Denition 2.5. A collection of sets S is called centered if the intersection of
every nite subfamily of S is innite.
We will use the following combinatorial principle, which is a consequence of
the Martin's Axiom.
p = c. For each centered collection A of fewer than c subsets of !, there is an
innite B ! such that B n A is nite for all A 2 A.
3. Results
Theorem 3.1. Let G be a topological group, jGj = and d(G) = < . Then
G is -resolvable.
Proof: Let D be a dense subset of G such that jD j = . Consider
H = hDi:
Then jH j = and H is dense in G. This implies G is -resolvable. Indeed,
consider the left coset space G=H which has cardinality and each coset gH 2
G=H is dense in G.
581
On resolvable spaces and groups
Consider a countable irresolvable topological space X . To the topological sum
of uncountably many copies of X add a point z . The neighborhoods of z are
all the summands with countably many exceptions. The resulting space Y is
uncountable Lindelof and irresolvable. However, for topological groups we have
the following result.
Theorem 3.2. Let G be a topological group, jGj = , G is -bounded and
> . Then G is resolvable.
Proof: We dene a strictly increasing sequence fH : 2 + g of subgroups of
G satisfying jH j for each < + .
If < + is a successor = + 1, choose K G n H , jK j = jj, and let
[
H = hK H i:
S
If < + is a limit, choose K G n 2 H , with jK j = jj, and put
H = hS [
[
2
H i:
ClearlySH H for each 2 . This completes our construction.
S Now we
set H = 2+ H G, and jH j > . We shall prove that H = <+ H is
resolvable. We dene the following sets
[
[
L=
(H n H );
2
limit
and
[
[
S=
(H n H ):
2
Successor
Obviously, H = L [ S and L \ S = ;. We claim that L and S are dense in H .
First, H is not discrete because it is -bounded and jH j > .
Suppose that S is not dense in H . Then there exists a non-empty open subset
U of H with U L. We have U = qV for some q 2 H , where V is a neighborhood
of e in H , so V = q,1 U . There exists a subset K H with jK j = < such
that
H = K V = K (q,1 U ):
Obviously, K H for some 2 + . We have q,1 2 H for some 2 + . Let
= maxf; g, and choose h 2 H+1 n H . There is k 2 K with h = kq,1 u,
u 2 U , and since k and q,1 are elements of H+1 , we get
u = qk,1 h 2 H+1 ;
which contradicts the assumption U L.
The same reasoning shows that L is dense in H . By Theorem 2.4 G is resolvable.
582
L.M.Villegas-Silva
Corollary 3.3. Every uncountable !-bounded group G is resolvable.
Proof: Choose a subset K G of cardinality @1 . Then dene
H = hK i:
The subgroup H has cardinality @1 and it is !-bounded. Now apply Theorem 3.2
to obtain the resolvability of H and therefore of G.
Now we deduce some results about homogeneous spaces which contain convergent sequences. This will enable us to establish some constraints concerning
irresolvable groups. A similar result has been noted independently by Comfort
and Garca-Ferreira [CG] and by Comfort, Masaveau and Zhou [CMZ].
Lemma 3.4. Let G be an innite countable homogeneous space which contains
a non-trivial convergent sequence. Then G is resolvable.
Proof: Let hxn j n 2 ! i G be a convergent sequence with the limit point
x, and Gg = hg1 ; g2 ; : : : i. For each gn 2 G, there exists a convergent sequence
Ag = hxi j i 2 !i, which has gn as a limit point.
Let B = hBi j i 2 !i be a family of innite subsets of G such that each Ag ,
n 2 !, occurs in this sequence innitely many times.
Now construct nite subsets En ; Fn of G by induction on n. Take y1 ; z1 2 B1 ,
y1 6= z1 and put E1 = fy1g, F1 = fz1 g. Suppose we have dened the sets
n
n
n
En,1 = fy1 ; : : : ; yn,1 g;
Fn,1 = fz1; : : : ; zn,1g:
Choose distinct elements yn ; zn of the set Bn n (En,1 [ Fn,1 ) and dene
En = En,1 [ fyng Fn = Fn,1 [ fzng:
This completes our construction. Now dene
1
[
F = Fn E = G n F:
n=1
Then G = E [ F and we claim that both E and F are dense in G. Indeed, let U
be an open set in G and gn 2 U . Almost all the points of Ag lie in U , so by the
construction U \ E 6= ; 6= U \ F . Thus G is resolvable.
Theorem 3.5. Let G be an homogeneous space, which contains a convergent
sequence. Then G is resolvable.
Proof: Let X = hxn : n 2 ! i be a convergent sequence with limit point x,
and S = X [ fxg. We construct inductively an innite sequence hSn : n 2 !i of
subspaces of S . Let S1 = S and for each pair of points xi ; xj of Sn,1 there exists
a homeomorphism hi;j : G ! G such that xi = hi;j (xj ). Let Kn,1 be the set of
all such homeomorphisms hi;j , for all i; j 2 !.
n
583
On resolvable spaces and groups
Case 1. The group G has cardinality !. This is given by Lemma 3.4.
Case 2. The group G is uncountable.
Let
[
Sn =
hi;j (Sn,1 )
h 2G ,1
and
[
S! = Sn :
n2!
Clearly S! is a countable homogeneous subspace which contains a convergent
sequence. By Theorem 3.4 S! is resolvable. For every
S g 2 G there exists a homeomorphism hg : G ! G with hg (x) = g. Thus G = g2G hg (S! ) and hence G is
resolvable by Theorem 2.5.
We now turn to irresolvable and maximal groups.
Theorem 3.6. (p = c) Let G be a countable irresolvable topological group. Then
w(G) = c.
Proof: Assume that w (G) < c. Consider a base F (e) of the lter of neighborhoods of the identity element with jF (e)j < c. The family F (e) has the nite
intersection property. By the principle P = c we get a set B G, jB j = @0 , such
that B converges to e. Therefore G is resolvable by Theorem 3.4, a contradiction.
i;j
n
Corollary 3.7. (p = c) If G is an irresolvable topological group, then w(G) c.
Corollary 3.8. (p = c) Let G be a maximal topological group. Then w(G) c.
4. Questions
We end with some open problems.
1. Does there exist an irresolvable topological group which is not maximal?
A topological space is extremally disconnected if for every open set U X , U ,
is open.
Louveau [L] has shown that there exists under CH an extremally disconnected
topological group. That group is resolvable. A natural question is therefore:
2. Does there exist an irresolvable topological group which is not extremally
disconnected?
It was proved in [CGvM] that every pseudocompact topological group is resolvable. In [GG] spaces weakly-pseudocompact were dened as those spaces with the
property of being G -dense in some of their compactications. It is then natural
to ask:
3. Is every weakly-pseudocompact group resolvable?
The proof of Theorem 3.2 does not give -resolvability of G.
584
L.M.Villegas-Silva
4. Is every uncountable !-bounded group -resolvable for some innite cardinal ? The referee has given an armative answer to this question for the case
= !.
5. Is it true that every -bounded group of cardinality greater than is resolvable? (See Theorem 3.3.)
I would like to express my gratitude to Professor M. Tkachenko and to Professor
S. Garca-Ferreira for his valuable advice on this work. I am also extremely
grateful to the referee for numerous suggestions and corrections which contributed
to the nal version of the paper.
References
[A]
Anderson D.R., On connected irresolvable Hausdor spaces, Proc. Amer. Math. Soc.
16 (1965), 463{466.
[Be]
Bell M.G., On the combinatorial principle P (c), Fund. Math. 114 (1981), 149{157.
[Bo]
Booth D., Ultralters on a countable set, Ann. Pure Appl. Logic (1970), 1{24.
[CF]
Comfort W.W., Feng L., The union of resolvable space is resolvable, preprint, 1993.
[CG]
Comfort W.W., Garca-Ferreira S., manuscript in preparation, 1900.
[CGvM] Comfort W.W., Gladdines H., Van Mill J., Proper pseudocompact subgroups of pseudocompact Abelian groups, preprint, 1993.
[CvM] Comfort W.W., Van Mill J., Groups with only resolvable topologies, preprint, 1993.
[CMZ] Comfort W.W., Masaveau O., Zhou H., Resolvability in topology and in topological
groups, Proc. Ninth (June 1993) Summer Topology Conference, Ann. New York Acad.
Sci., to appear.
[GG]
Garca-Ferreira S., Garca-Maynez A., On weakly pseudocompact spaces, Houston J.
Math. 20 (1994), 145{159.
[G]
Guran I.I., On topological groups close to being Lindelof, Soviet Math. Dokl. 23
(1981), 173{175.
[H]
Hewitt E., A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309{333.
[L]
Louveau A., Sur un article de S. Sirota, Bull. Sci. Math. (2) 96 (1972), 3{7.
[M]
Malykhin V.I., Extremally disconnected and similar groups, Soviet Math. Dokl. 16
(1975), 21{25.
[P]
Padmavally K., An example of a connected irresolvable Hausdor spaces, Duke Math.
J. 20 (1953), 513{520.
Universidad Autonoma Metropolitana Iztapalapa, Av. Michoacan y La Purisima,
Iztapalapa, D.F., C.P. 09340, Mexico
E-mail : [email protected] xanum.uam.mx
(Received August 2, 1994, revised Jamuary 18, 1995)